For the characterization of MI spaces, the range function plays a crucial role. The history of the range function started from the work of Helson [1]} and then widely been used for the characterization of translation invariant spaces and multiplication invariant spaces [2]}, [3]}, [4]}.
A range function on \(X\)  is a mapping
\(J:X\rightarrow \lbrace \text{closed subspaces of} ~\mathcal {H}\rbrace \) .
Further, we say \(J\)  is measurable if for any \(u,v\in \mathcal {H}\)  the mapping
\(x\mapsto \langle P_J(x)u,v\rangle \)  is measurable on \(X\) , where for \(x\in X\) , the orthogonal projection \(P_{J} (x): \mathcal {H} \rightarrow \mathcal {H}\)  projects onto \(J(x)\) .
We can associate a closed subspace \(V_J\)  with every projection-valued map \(J\)  as follows:
\(V_J:=\left\lbrace \varphi \in L^2(X;\mathcal {H}):\varphi (x)\in J(x)~\mbox{for}~ a.e.~ x\in X\right\rbrace .\) 
